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Unformatted text preview: Lecture 4 More on ReCasting Problems in Terms of Random Variables Example 1 (13.44 on p.427) The problem statement is as follows: In a study of the effectiveness of certain exercises in weight reduction, a group of 16 persons engaged in these exercises for one month and showed the following results: Table 1. Measured weights (lbs.) before and after exercise program Person Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Before 211 180 171 214 182 194 160 182 172 155 185 167 203 181 245 146 After 198 173 172 209 179 192 161 182 166 154 181 164 201 175 233 142 The numbers in the table did not arise spontaneously. They were the result of actions. Since there are 32 numbers, there were 32 actions. There are 32 numbers because for each of 16 persons, two actions were taken. Q: What are the two actions that were taken for each of the 16 persons? A: X = measuring a persons weight before the program ; Y = measuring that persons weight after. Hence, it is natural to let X k ( Y k ) denote the act of measuring the k th persons wgt. before (after) the program. NOTATION: There are two ways to denote the 32 random variables in a concise manner: (i) as two collections (or sets): 16 1 } { = k k X & 16 1 } { = k k Y . (ii) as two vectors (or, arrays): X = ] , , , [ 16 2 1 X X X & Y = ] , , , [ 16 2 1 Y Y Y . The advantage of the form (ii) is that we can then use matrix operations. For example, a scatter plot of the x / y data is shown below. 140 160 180 200 220 240 260 140 150 160 170 180 190 200 210 220 230 240 x = weight before program y = weight after program Figure 1. Scatter plot of the measured weights of the 16 persons before versus after the program. The scatter plot suggests that for any arbitrarily chosen person who might enroll in the program, one might be able to predict what the act of measuring his/her weight after the program ( Y ) will be in relation to the measurement of his/her weight before the program ( X ). Let Y t denote the act of predicting what the persons weight will be after the program. Since the scatter plot has a relatively linear appearance, it is reasonable to use a linear prediction model : b X m Y + = . (1) The model (1) is the slopeintercept form for a straight line with slope...
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This note was uploaded on 11/01/2009 for the course EE 447 taught by Professor Dontknow during the Spring '09 term at Iowa State.
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